[GoLUG] cal 9 1752 and the Gregorian calendar revision

Omer Zak w1 at zak.co.il
Fri Mar 7 00:28:52 EST 2025 

I decided to compare the Gregorian calendar to the Jewish calendar.
Due to my ignorance and laziness, I had ChatGPT make the actual
calculations and write an essay about this subject.

It did not make precise calculations but took percentages of each type
of Jewish year (of which there are 6) so I accept the conclusions with
a grain of salt.

The muslim calendar has a built-in mechanism of adjusting to actual
year length whatever it is - the new moon needs to be actually observed
for a new month to be declared.


================> Beginning of essay


The Precision of the Jewish Calendar and Its Comparison to the
Gregorian Calendar

Introduction
Calendars play a vital role in human civilization, organizing time for
religious, agricultural, and administrative purposes. Among the many
calendar systems developed, the Jewish calendar and the Gregorian
calendar stand out as two of the most refined. While the Jewish
calendar is a lunisolar system designed to keep religious observances
in alignment with both lunar months and the solar year, the Gregorian
calendar is a purely solar calendar optimized for long-term precision
in tracking the Earth’s revolution around the Sun. This essay explores
the precise calculations behind the Jewish calendar, its long-term
accuracy, and how it compares to the Gregorian calendar in terms of
drift and long-term sustainability.

Understanding the Jewish Calendar
The Jewish calendar is based on a 19-year Metonic cycle, in which 7 of
the 19 years are leap years that include an extra month, Adar II, to
keep the calendar in sync with the solar year. Unlike purely lunar
calendars, this system ensures that Jewish holidays remain within their
intended seasons over long periods.

However, the Jewish calendar is more complex than just adding 7 leap
months per 19 years. Each year can have one of six possible lengths,
due to adjustments made to keep the months aligned with both lunar
phases and seasonal requirements:
- Regular years (12 months):
  - Deficient (353 days)
  - Regular (354 days)
  - Complete (355 days)
- Leap years (13 months):
  - Deficient (383 days)
  - Regular (384 days)
  - Complete (385 days)

These variations arise due to constraints in the Hebrew calendar, such
as postponement rules to avoid certain holidays falling on specific
weekdays. Over a full 19-year cycle, the Jewish calendar closely
approximates the solar year, but minor differences accumulate over
millennia.

Calculating the Drift of the Jewish Calendar
A precise analysis of the Jewish calendar requires considering the
real-world statistical distribution of these different year lengths.
Based on historical records and halachic principles, we can estimate
the frequencies of each year type:
- Regular years (12 months):
  - 353 days (20.5%)
  - 354 days (46.0%)
  - 355 days (33.5%)
- Leap years (13 months):
  - 383 days (25.0%)
  - 384 days (50.0%)
  - 385 days (25.0%)

Using these proportions, the average length of a Jewish year is
calculated as follows:

(12 × 354.37) + (7 × 384.37) / 19 = 365.2424 days

This result shows that the Jewish calendar's average year length is
365.2424 days, which is remarkably close to the true tropical year of
365.2422 days. The small difference of +0.0002 days per year (≈ 17.28
seconds per year) means that the Jewish calendar drifts by 1 day every
~5,000 years. This is significantly more precise than earlier
estimates, which suggested a drift of 1 day every ~216 years based on
rougher approximations.

The Gregorian Calendar’s Precision
The Gregorian calendar, introduced by Pope Gregory XIII in 1582, was
designed to correct the drift present in the older Julian calendar,
which overestimated the solar year and accumulated a 10-day error by
the 16th century. The Gregorian reform refined the leap year rule:
- A year is a leap year if divisible by 4.
- Exception: If the year is divisible by 100, it is not a leap year
unless it is also divisible by 400.

This rule results in an average Gregorian year length:

(365 × 3 + 366) / 4 - 1/100 + 1/400 = 365.2425 days

The Gregorian calendar’s year length of 365.2425 days is only 0.0003
days longer than the true tropical year, meaning it drifts by 1 day
every ~3,333 years.

Comparing the Long-Term Drift of Both Calendars
A direct comparison of drift rates highlights how close both calendars
are in long-term precision:

| Calendar | Average Year Length (days) | Drift per Year | 1-Day Drift
Time |
|----------|--------------------|----------------|------------------|
| Jewish Calendar (revised estimate) | 365.2424 | +0.0002 days | ~5,000
years |
| Gregorian Calendar | 365.2425 | +0.0003 days | ~3,333 years |

Surprisingly, the Jewish calendar is actually more accurate than
previously thought, with a slightly better drift rate than the
Gregorian calendar. However, both are exceptionally precise and remain
accurate over thousands of years.

Implications and Potential Corrections
1. Jewish Calendar Adjustments:
   - Unlike the Gregorian calendar, which includes leap year rules to
maintain accuracy, the Jewish calendar currently does not have built-in
long-term corrections.
   - If drift became noticeable over thousands of years, a simple
adjustment (such as skipping a leap year once every ~5,000 years) could
maintain alignment.

2. Gregorian Calendar Refinements:
   - The Gregorian leap year system is an approximation, and minor
future refinements could be considered.
   - Some scholars suggest a further correction, such as omitting a
leap day once every ~3,300 years, to maintain perfect alignment.

Conclusion
The Jewish calendar’s Metonic cycle provides an extremely close match
to the solar year, with an updated drift estimate of 1 day per 5,000
years—far more accurate than rough estimates previously suggested. The
Gregorian calendar, with a drift of 1 day per 3,333 years, remains the
most widely used and refined system for tracking the solar year.

While both calendars are designed for different purposes—the Jewish
calendar balancing lunar and solar cycles for religious observances and
the Gregorian calendar ensuring secular timekeeping accuracy—they both
demonstrate remarkable precision. Future adjustments may be required
over millennia, but for all practical purposes, these calendars serve
humanity’s needs exceptionally well.

Thus, whether one follows the Jewish calendar’s sacred cycles or the
Gregorian calendar’s astronomical precision, both stand as testaments
to humanity’s quest to master timekeeping.


================> End of essay


--- Omer Zak



On Thu, 2025-03-06 at 20:21 -0800, Kyle Terrien wrote:
> During GoLUG yesterday, I dared everyone to run ‘cal 9 1752’.  No,
> this is not a prank or computer cracking trick.  This is the month
> that England and her colonies switched from the Julian Calendar to
> the
> Gregorian Calendar.  11 days were skipped to catch up with the
> Gregorian Calendar.
> 
>    September 1752
>  S  M Tu  W Th  F  S
>        1  2 14 15 16
> 17 18 19 20 21 22 23
> 24 25 26 27 28 29 30
> 
> I did a little bit of digging and went down a historical rabbit hole.
> 
> The Gregorian Calendar[1] is a calendar revision promulgated by Pope
> Gregory XIII in 1582.  It replaced the Julian calendar, which has
> leap
> years every 4 years.  In the Gregorian system, years divisible by 100
> are not leap years unless they are also divisible by 400, for a grand
> cycle of 400 years.  This extra rule was invented to align the
> calendar better with the summer solstice and make the calculation of
> Easter more reliable.
> 
> How does one calculate Easter?  The formula is surprisingly
> complex[2], despite the formal definition of “Sunday after the Jewish
> Passover” being fixed at the Council of Nicaea (AD 325).  If you use
> Emacs, take a look at holidays.el for an algorithmic implementation.
> 
> Anyway, the purpose of the Gregorian system was to better align the
> calendar with the solstice and lunar cycles to make computing the day
> of Easter more reliable.
> 
> So, now onto what happened after 1582...
> 
> Different countries adopted the Gregorian calendar at different
> times[3].  There were a couple centuries where the same day had
> different designations in different countries.  Sometimes dates were
> labeled “Old Style” (O.S.) and “New Style” (N.S.).
> 
> But, if you look at the timeline in [3], you will notice a few
> patterns.
> 
> The first countries that switched to the Gregorian system were the
> Catholic countries.  They switched almost immediately.  This
> shouldn’t
> be a surprise because Pope Gregory XIII was Catholic, and the
> Catholics have a strong hierarchy where Church state exists above the
> civil state.  So, adoption was a no-brainer.
> 
> Then came the Protestant countries, Germany and Great Britain (which
> was dragged into Protestantism by Henry VIII, who wanted to divorce
> his wife---story for another time).  British politics is where we get
> the Sep 1752 hard-coded date in cal.
> 
> Then, after the Protestant countries, we have the Eastern Orthodox
> countries (Russia, Bulgaria, Greece, etc.).  Here is where some
> historical context is required.
> 
> The Eastern Orthodox religion goes back to the Cerulean Schism of
> 1054, better known as the East-West Schism or “Great Schism”[4].  In
> the Cerulean Schism, the eastern Churches broke unity with the
> Catholic Church in the west (or vice versa depending on who you
> ask---I’m trying to be diplomatic here).
> 
> Importantly, 1054 was 500 years before the Gregorian calendar, so the
> Orthodox used the Julian calendar just like all the Catholics of
> 1054.
> The Orthodox were very *very* concerned with tradition, so they kept
> using the Julian calendar even after the Gregorian revision.  So,
> some
> of the last countries to switch in the chart in [3] are Orthodox
> countries.
> 
> But wait, it gets even *better*.  In the 1920s, a Serbian scientist
> (Orthodox) created a Revised Julian calendar[5].  This is known as
> the
> “new calendar” within Orthodox circles, and this is the one that the
> “Old Calendarists”[6] to which I alluded are adamantly against.
> However, the Revised Julian calendar has a rule different from the
> Gregorian Calendar:
> 
> > years evenly divisible by 100 are not leap years, except that years
> > with remainders of 200 or 600 when divided by 900 remain leap
> > years[5]
> 
> It just so happens this coincides day-for-day with the Gregorian
> calendar only from AD 1700 to AD 2800.
> 
> And *that* is one of the craziest things I have learned in a while.
> The Western countries have one calendar, and the Eastern countries
> have another.  They just happen to coincide for several centuries,
> partly out of design.  After 28 February 2800, the East and West will
> be off by one day, and someone will need to do something about it
> again.
> 
> --Kyle
> 
> 
> [1]: https://en.wikipedia.org/wiki/Gregorian_calendar
> [2]: https://en.wikipedia.org/wiki/Date_of_Easter
> [3]:
> https://en.wikipedia.org/wiki/Adoption_of_the_Gregorian_calendar#Timeline
> [4]: https://en.wikipedia.org/wiki/East%E2%80%93West_Schism
> [5]: https://en.wikipedia.org/wiki/Revised_Julian_calendar
> [6]: https://en.wikipedia.org/wiki/Old_Calendarists


-- 
cal 09 1752
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